Source for that first claim?
I thought it was only National Parks.
Here are some references. First is the memorandum that established the drone bane in National Park Service Units.
The important text is:
As used in this Policy Memorandum and its exhibits, the term “park” means any unit of the National Park System.
Next, to understand what "all units" means, you can see the following at this link:
Designations of National Park System Units - Golden Gate National Recreation Area (U.S. National Park Service)
Designations of National Park System Units
The numerous designations within the National Park System sometime confuse visitors. The names are created in the Congressional legislation authorizing the sites or by the president, who proclaims "national monuments" under the Antiquities Act of 1906. Many names are descriptive—lakeshores, seashores, battlefields—but others cannot be neatly categorized because of the diversity of resources within them. In 1970, Congress elaborated on the 1916 National Park Service Organic Act, saying all units of the system have equal legal standing in a national system.
National Park
These are generally large natural places having a wide variety of attributes, at times including significant historic assets. Hunting, mining and consumptive activities are not authorized.
National Monument
The Antiquities Act of 1906 authorized the President to declare by public proclamation landmarks, structures, and other objects of historic or scientific interest situated on lands owned or controlled by the government to be national monuments.
National Preserve
National preserves are areas having characteristics associated with national parks, but in which Congress has permitted continued public hunting, trapping, oil/gas exploration and extraction. Many existing national preserves, without sport hunting, would qualify for national park designation.
National Historic Site
Usually, a national historic site contains a single historical feature that was directly associated with its subject. Derived from the Historic Sites Act of 1935, a number of historic sites were established by secretaries of the Interior, but most have been authorized by acts of Congress.
National Historical Park
This designation generally applies to historic parks that extend beyond single properties or buildings.
National Memorial
A national memorial is commemorative of a historic person or episode; it need not occupy a site historically connected with its subject.
National Battlefield
This general title includes national battlefield, national battlefield park, national battlefield site, and national military park. In 1958, an NPS committee recommended national battlefield as the single title for all such park lands.
National Cemetery
There are presently 14 national cemeteries in the National Park System, all of which are administered in conjunction with an associated unit and are not accounted for separately.
National Recreation Area
Twelve NRAs in the system are centered on large reservoirs and emphasize water-based recreation. Five other NRAs are located near major population centers. Such urban parks combine scarce open spaces with the preservation of significant historic resources and important natural areas in location that can provide outdoor recreation for large numbers of people.
National Seashore
Ten national seashores have been established on the Atlantic, Gulf and Pacific coasts; some are developed and some relatively primitive. Hunting is allowed at many of these sites.
National Lakeshore
National lakeshores, all on the GreatLakes, closely parallel the seashores in character and use.
National River
There are several variations to this category: national river and recreation area, national scenic river, wild river, etc. The first was authorized in 1964 and others were established following passage of the Wild and Scenic Rivers Act of 1968.
National Parkway
The title parkway refers to a roadway and the parkland paralleling the roadway. All were intended for scenic motoring along a protected corridor and often connect cultural sites.
National Trail
National scenic trails and national historic trails are the titles given to these linear parklands (over 3,600 miles) authorized under the National Trails System Act of 1968.
Affiliated Areas
In an Act of August 18, 1970, the National Park System was defined in law as, "any area of land and water now or hereafter administered by the Secretary of the Interior through the National Park Service for park, monument, historic, parkway, recreational or other purposes."The Affiliated Areas comprise a variety of locations in the United States and Canada that preserve significant properties outside the National Park System. Some of these have been recognized by Acts of Congress, others have been designated national historic sites by the Secretary of the Interior under authority of the Historic Sites Act of 1935. All draw on technical or financial aid from the National Park Service.
Other Designations
Some units of the National Park System bear unique titles or combinations of titles, like the White House and Prince William Forest Park.
If you go to the web sites of these units, you can see rules specific to the unit and all that I have looked at contain the drone ban.
